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7. Let e(t) = r(t)-y(t). Compute the steady state values y(oo) and e(oo) when both r(t) and d(t) are unit step functions (at the same time). There are several ways to approach this problem, but one way is to use the property of superposition. R(s) s+4 D(s) 1 8³+28² +8 Y(s)

7. Let e(t) = r(t)-y(t). Compute the steady state values y(oo) and e(oo) when both r(t) and d(t) are unit step functions (at the same time). There are several ways to approach this problem, but one way is to use the property of superposition. R(s) s+4 D(s) 1 8³+28² +8 Y(s)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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The computed steady state values are given, but please show the work needed to arrive at our solution.

**Problem Statement:** 7. Let \( e(t) = r(t) - y(t) \). Compute the steady state values \( y(\infty) \) and \( e(\infty) \) when both \( r(t) \) and \( d(t) \) are unit step functions (at the same time). There are several ways to approach this problem, but one way is to use the property of superposition. **Block Diagram Explanation:** The diagram is a control system block diagram consisting of the following components: 1. **Input \( R(s) \):** The Laplace transform of the reference input signal. 2. **Summing Junction (−):** - **Inputs:** \( R(s) \) and feedback from the output \( Y(s) \). - **Operation:** The output of the summing junction is \( R(s) - Y(s) \). 3. **Block \( s + 4 \):** This block represents a transfer function with a gain of \( s + 4 \). 4. **Another Summing Junction (+):** - **Inputs:** Output from the \( s+4 \) block and disturbance \( D(s) \). - **Operation:** Adds the output of the \( s+4 \) block and \( D(s) \). 5. **Transfer Function Block:** \( \frac{1}{s^2 + 2s + 8} \) - This block processes the output of the sum from the previous summing junction. 6. **Output \( Y(s) \):** The Laplace transform of the system output. **Objective:** Find the steady state values of \( y(t) \) and \( e(t) = r(t) - y(t) \) when the inputs \( r(t) \) and \( d(t) \) are unit step functions. The superposition principle is suggested as a method to solve this problem.
Transcribed Image Text:**Problem Statement:** 7. Let \( e(t) = r(t) - y(t) \). Compute the steady state values \( y(\infty) \) and \( e(\infty) \) when both \( r(t) \) and \( d(t) \) are unit step functions (at the same time). There are several ways to approach this problem, but one way is to use the property of superposition. **Block Diagram Explanation:** The diagram is a control system block diagram consisting of the following components: 1. **Input \( R(s) \):** The Laplace transform of the reference input signal. 2. **Summing Junction (−):** - **Inputs:** \( R(s) \) and feedback from the output \( Y(s) \). - **Operation:** The output of the summing junction is \( R(s) - Y(s) \). 3. **Block \( s + 4 \):** This block represents a transfer function with a gain of \( s + 4 \). 4. **Another Summing Junction (+):** - **Inputs:** Output from the \( s+4 \) block and disturbance \( D(s) \). - **Operation:** Adds the output of the \( s+4 \) block and \( D(s) \). 5. **Transfer Function Block:** \( \frac{1}{s^2 + 2s + 8} \) - This block processes the output of the sum from the previous summing junction. 6. **Output \( Y(s) \):** The Laplace transform of the system output. **Objective:** Find the steady state values of \( y(t) \) and \( e(t) = r(t) - y(t) \) when the inputs \( r(t) \) and \( d(t) \) are unit step functions. The superposition principle is suggested as a method to solve this problem.
7. \( y(\infty) = 5/4 \) and \( e(\infty) = -1/4 \). This text appears to be part of a mathematical or engineering-related problem, possibly concerning limits or steady-state values of certain functions or systems as they approach infinity.
Transcribed Image Text:7. \( y(\infty) = 5/4 \) and \( e(\infty) = -1/4 \). This text appears to be part of a mathematical or engineering-related problem, possibly concerning limits or steady-state values of certain functions or systems as they approach infinity.
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